- Pentagonal number
A

**pentagonal number**is afigurate number that extends the concept of triangular andsquare number s to thepentagon , but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The "n"th pentagonal number "p"_{n}is the number of "distinct" dots in a pattern of dots consisting of the "outlines" of regular pentagons whose sides contain 1 to n dots, overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside..."p"

_{n}is given by the formula::$p\_n\; =\; frac\{n(3n-1)\}2$

for "n" ≥ 1. The first few pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001 OEIS|id=A000326.

The "n"th pentagonal number is one third of the 3"n"-1th

triangular number .**Generalized pentagonal numbers**are obtained from the formula given above, but with "n" taking values in the sequence 0, 1, -1, 2, -2, 3, -3, 4..., producing the sequence:0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027... OEIS|id=A001318.

Generalized pentagonal numbers are important to

Euler 's theory of partitions, as expressed in hispentagonal number theorem .The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.

Pentagonal numbers should not be confused with

centered pentagonal number s.**Tests for pentagonal numbers**One can test whether a positive integer "x" is a (non-generalized) pentagonal number by computing

:$n\; =\; frac\{sqrt\{24x+1\}\; +\; 1\}\{6\}.$

If "n" is a

natural number , then "x" is the "n"th pentagonal number. If "n" is not a natural number, then "x" is not pentagonal.

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